Difference between revisions of "Courant theorem"
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Let $\{D_n\}$ be a sequence of nested simply-connected domains in the complex $z$-plane, $\overline{D_{n+1}}\subset D_n$, which converges to its kernel $D_{z_0}$ relative to some point $z_0$; the set $D_{z_0}$ is assumed to be bounded by a Jordan curve. Then the sequence of functions $\{w=f_n(z)\}$ which map $D_n$ conformally onto the disc $\Delta=\{w:|w|<1\}$, $f_n(z_0)=0$, $f'_n(z_0)>0$, is uniformly convergent in the closed domain $\overline{D_{z_0}}$ to the function $w=f(z)$ which maps $D_{z_0}$ conformally onto $\Delta$, moreover $f(z_0)=0$, $f'(z_0)>0$. | Let $\{D_n\}$ be a sequence of nested simply-connected domains in the complex $z$-plane, $\overline{D_{n+1}}\subset D_n$, which converges to its kernel $D_{z_0}$ relative to some point $z_0$; the set $D_{z_0}$ is assumed to be bounded by a Jordan curve. Then the sequence of functions $\{w=f_n(z)\}$ which map $D_n$ conformally onto the disc $\Delta=\{w:|w|<1\}$, $f_n(z_0)=0$, $f'_n(z_0)>0$, is uniformly convergent in the closed domain $\overline{D_{z_0}}$ to the function $w=f(z)$ which maps $D_{z_0}$ conformally onto $\Delta$, moreover $f(z_0)=0$, $f'(z_0)>0$. | ||
| − | This theorem, due to R. Courant , is an extension of the [[Carathéodory theorem|Carathéodory theorem]]. | + | This theorem, due to R. Courant [[#References|[1]]], is an extension of the [[Carathéodory theorem|Carathéodory theorem]]. |
====References==== | ====References==== | ||
Latest revision as of 21:14, 22 December 2018
on conformal mapping of domains with variable boundaries
Let $\{D_n\}$ be a sequence of nested simply-connected domains in the complex $z$-plane, $\overline{D_{n+1}}\subset D_n$, which converges to its kernel $D_{z_0}$ relative to some point $z_0$; the set $D_{z_0}$ is assumed to be bounded by a Jordan curve. Then the sequence of functions $\{w=f_n(z)\}$ which map $D_n$ conformally onto the disc $\Delta=\{w:|w|<1\}$, $f_n(z_0)=0$, $f'_n(z_0)>0$, is uniformly convergent in the closed domain $\overline{D_{z_0}}$ to the function $w=f(z)$ which maps $D_{z_0}$ conformally onto $\Delta$, moreover $f(z_0)=0$, $f'(z_0)>0$.
This theorem, due to R. Courant [1], is an extension of the Carathéodory theorem.
References
| [1a] | R. Courant, Gott. Nachr. (1914) pp. 101–109 |
| [1b] | R. Courant, Gott. Nachr. (1922) pp. 69–70 |
| [2] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian) |
Comments
Cf. Carathéodory theorem for the definition of "kernel of a sequence of domains" .
Courant theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Courant_theorem&oldid=43540